Dini's surface

In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere.^[1] It is named after Ulisse Dini^[2] and described by the following parametric equations:^[3]

${\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}}$

Dini's surface with 0 ≤ u ≤ 4π and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2.

Dini's surface plotted with adjustable parameters by Wolfram Mathematica program

Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4π and 0<v<1.

Another description is a generalized helicoid constructed from the tractrix.^[4]

See also

• Breather surface
• Kuen surface